7. Lid Driven Cavity Flow

In this notebook we will simulate the flow with a top moving wall. To demonstrate this, we will solve for internal flow in a square cavity by enforcing no-slip at the walls.

using ViscousFlow

using Plots

Problem specification

Take $Re=100$ for example:

Re = 100 # Reynolds number

100

Discretization

Note that the rectangle function used for making the cavity shape requires a specified half length. The immersed boundary projection method for internal flow requires the size of the domain to be at least a step size greater at the boundaries (i.e. halflength + Δx).

Δt,Δx = setstepsizes(Re,gridRe=1.0)
halflength=0.5 # to make rectangle with side length of 1
domain_lim=halflength+1.01*Δx; # 1.01 is just an abitrary factor chosen to be greater than 1
xlim, ylim = (-domain_lim,domain_lim),(-domain_lim,domain_lim)

((-0.50505, 0.50505), (-0.50505, 0.50505))

Cavity Geometry

A square cavity can be created using the Rectangle() function with the half length defined above. The shifted=true argument ensures that points are not placed at the corners, where they have ill-defined normal vectors.

body = Rectangle(halflength,halflength,1.5*Δx,shifted=true)
plot(body,fillrange=nothing)

Boundary Condition at the moving wall

Assign velocity to the top boundary. The LidDrivenCavity() function can be used to specify the velocity value at the top wall. Note : Non-dimensional velocity = 1

m = ViscousFlow.LidDrivenCavity(1.0)

ViscousFlow.LidDrivenCavity(1.0)

Construct the system structure

The last two input flow_side and static_points must specified so the default setting in the NavierStokes() function can be overwritten.

static_points is set to true because the cavity wall points are not actually moving.

sys = NavierStokes(Re,Δx,xlim,ylim,Δt,body,m,flow_side = InternalFlow,static_points = true)

Internal flow Navier-Stokes system on a grid of size 208 x 208 and 528 static immersed points
   Static freestream = (0.0, 0.0)
   1 body

Initialize

u0 = newstate(sys)

(Dual nodes in a (nx = 208, ny = 208) cell grid of type Float64 data
  Number of Dual nodes: (nx = 208, ny = 208), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])

Set up integrator

tspan = (0.0,10.0)
integrator = init(u0,tspan,sys)

t: 0.0
u: (Dual nodes in a (nx = 208, ny = 208) cell grid of type Float64 data
  Number of Dual nodes: (nx = 208, ny = 208), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])

Solve

step!(integrator,10)

Examine

Plot the vorticity and streamlines

plot(
plot(vorticity(integrator),sys,title="Vorticity",clim=(-10,10),color=:turbo,linewidth=1.5,ylim=ylim,levels=-6:0.5:5),
plot(streamfunction(integrator),sys,title="Streamfunction",color=:black,levels=vcat(0.009:0.01:0.11,0.1145,0.11468,0.11477),ylim=ylim)
   )

Make a movie:

sol = integrator.sol;
@gif for (u,t) in zip(sol.u,sol.t)
    plot(vorticity(u,sys,t),sys,clim=(-10,10),levels=range(-10,10,length=30),color=:turbo)
end every 5

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